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Jerard Puckett mentioned in The Pod Bay yesterday that Shuttles did S-Curve maneuvers back and forth across their reentry trajectory to shed speed. After a bit of looking, I found some reference to this, but not much. In the Kerbal Space Program Forum this thread mentioned it:

Also, space shuttles would re-enter in a rolled attitude to steepen their trajectory (i.e. the wing's lift was directed sideways not up). The roll would result in a cross-track velocity component but roll reversal maneuvers were used to "sway" the re-entry trajectory back and forth across the target landing site.

And they were discussing the photo below, of Atlantis starting reentry, I guess taken from the ISS, which shows how the maneuver looks, seen from behind and above. Atlantis is the bright spot, engulfed in plasma, doing a turn:

Shuttle Atlantis reentry trajectory as seen from ISS

How much of their velocity did the Shuttles shed this way? Was it a critical part of the reentry process?

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    $\begingroup$ It is uncited, and doesn't really directly answer your question, but en.wikipedia.org/wiki/Space_shuttle#Re-entry_and_landing does say that "In a straight line, its 40-degree nose-up attitude would cause the descent angle to flatten-out, or even rise. The vehicle therefore performed a series of four steep S-shaped banking turns, each lasting several minutes, at up to 70 degrees of bank, while still maintaining the 40-degree angle of attack. In this way it dissipated speed sideways rather than upwards." If you get a good answer with citations, you may want to update Wikipedia. $\endgroup$ – a CVn Apr 8 '16 at 20:23
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It's a bit of a misconception that the shuttle used S-turns to slow down. To quote the Entry, TAEM, and Approach/Landing Guidance Workbook by the United Space Alliance:

The next time you hear someone talk about the shuttle doing roll reversals to bleed off energy, do not listen. The shuttle does roll reversals because it has a very small alpha envelope.

(sorry, I don't have a public link for the workbook, but I know it's available with an L2 subscription on nasaspaceflight.com)

What it's saying is that the ideal reentry strategy is to simply point the nose at the landing site and control how fast you're dissipating energy by controlling angle of attack (alpha). A higher alpha will result in more drag, and it's easy to make quick adjustments to get the level of drag you need.

However, in order keep the orbiter both controllable and protected by the thermal protection system, it had to be flown at a particular alpha (40° for most of entry) with no more than 3° of variation. This meant they had to use different means to control drag, and the way they did it was by controlling the lift vector.

A steeper bank angle meant that the lift vector would be more sideways than up, which would cause the shuttle to descend quicker, and, as you descend, the air gets thicker, which causes more drag. A shallower bank angle will slow your descent and keep you in thinner air for longer, which minimizes your drag.

But there's a problem with using bank... it starts to turn you off course. So the solution is to use roll reversals (aka s-turns) to keep you pointed towards the landing site.

Now, to sort of answer the literal question you asked (how much entry speed was bled off during the s-turns): most of it. The first roll would occur at nearly orbital velocity, and entry guidance ended at 2500 feet per second (Earth-relative velocity). After entry guidance ended, TAEM (Terminal Area Energy Management) guidance began, which primarily used alpha for energy management. Once the orbiter was subsonic, the speedbrake was also used for energy management.

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